metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊8M4(2), Dic5⋊9M4(2), C20⋊C8⋊15C2, C4⋊2(C22.F5), C22.7(C4⋊F5), (C22×C4).22F5, C23.45(C2×F5), Dic5⋊C8⋊7C2, C5⋊3(C4⋊M4(2)), (C22×C20).24C4, (C4×Dic5).38C4, Dic5.37(C2×D4), Dic5.19(C2×Q8), (C2×Dic5).38Q8, Dic5.36(C4⋊C4), (C2×Dic5).180D4, C10.28(C2×M4(2)), C22.87(C22×F5), C2.18(D5⋊M4(2)), (C22×Dic5).34C4, (C2×Dic5).349C23, (C4×Dic5).348C22, (C22×Dic5).277C22, C2.22(C2×C4⋊F5), C10.21(C2×C4⋊C4), (C2×C5⋊C8).8C22, (C2×C4).144(C2×F5), (C2×C4×Dic5).47C2, (C2×C10).27(C4⋊C4), (C2×C20).111(C2×C4), C2.7(C2×C22.F5), (C2×C22.F5).5C2, (C22×C10).65(C2×C4), (C2×C10).65(C22×C4), (C2×Dic5).187(C2×C4), SmallGroup(320,1096)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 378 in 126 conjugacy classes, 58 normal (32 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×12], C23, C10 [×3], C10 [×2], C42 [×4], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×2], Dic5 [×2], Dic5 [×4], Dic5, C20 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×2], C4⋊C8 [×4], C2×C42, C2×M4(2) [×2], C5⋊C8 [×4], C2×Dic5 [×4], C2×Dic5 [×4], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×C10, C4⋊M4(2), C4×Dic5 [×4], C2×C5⋊C8 [×4], C22.F5 [×4], C22×Dic5 [×2], C22×C20, C20⋊C8 [×2], Dic5⋊C8 [×2], C2×C4×Dic5, C2×C22.F5 [×2], C20⋊8M4(2)
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C2×M4(2) [×2], C2×F5 [×3], C4⋊M4(2), C4⋊F5 [×2], C22.F5 [×2], C22×F5, D5⋊M4(2), C2×C4⋊F5, C2×C22.F5, C20⋊8M4(2)
Generators and relations
G = < a,b,c | a20=b8=c2=1, bab-1=a3, ac=ca, cbc=b5 >
►On 160 points
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(1 61 46 104 33 100 126 145)(2 68 55 107 34 87 135 148)(3 75 44 110 35 94 124 151)(4 62 53 113 36 81 133 154)(5 69 42 116 37 88 122 157)(6 76 51 119 38 95 131 160)(7 63 60 102 39 82 140 143)(8 70 49 105 40 89 129 146)(9 77 58 108 21 96 138 149)(10 64 47 111 22 83 127 152)(11 71 56 114 23 90 136 155)(12 78 45 117 24 97 125 158)(13 65 54 120 25 84 134 141)(14 72 43 103 26 91 123 144)(15 79 52 106 27 98 132 147)(16 66 41 109 28 85 121 150)(17 73 50 112 29 92 130 153)(18 80 59 115 30 99 139 156)(19 67 48 118 31 86 128 159)(20 74 57 101 32 93 137 142)
(61 100)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)(79 98)(80 99)(101 142)(102 143)(103 144)(104 145)(105 146)(106 147)(107 148)(108 149)(109 150)(110 151)(111 152)(112 153)(113 154)(114 155)(115 156)(116 157)(117 158)(118 159)(119 160)(120 141)
G:=sub<Sym(160)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,61,46,104,33,100,126,145)(2,68,55,107,34,87,135,148)(3,75,44,110,35,94,124,151)(4,62,53,113,36,81,133,154)(5,69,42,116,37,88,122,157)(6,76,51,119,38,95,131,160)(7,63,60,102,39,82,140,143)(8,70,49,105,40,89,129,146)(9,77,58,108,21,96,138,149)(10,64,47,111,22,83,127,152)(11,71,56,114,23,90,136,155)(12,78,45,117,24,97,125,158)(13,65,54,120,25,84,134,141)(14,72,43,103,26,91,123,144)(15,79,52,106,27,98,132,147)(16,66,41,109,28,85,121,150)(17,73,50,112,29,92,130,153)(18,80,59,115,30,99,139,156)(19,67,48,118,31,86,128,159)(20,74,57,101,32,93,137,142), (61,100)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,99)(101,142)(102,143)(103,144)(104,145)(105,146)(106,147)(107,148)(108,149)(109,150)(110,151)(111,152)(112,153)(113,154)(114,155)(115,156)(116,157)(117,158)(118,159)(119,160)(120,141)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,61,46,104,33,100,126,145)(2,68,55,107,34,87,135,148)(3,75,44,110,35,94,124,151)(4,62,53,113,36,81,133,154)(5,69,42,116,37,88,122,157)(6,76,51,119,38,95,131,160)(7,63,60,102,39,82,140,143)(8,70,49,105,40,89,129,146)(9,77,58,108,21,96,138,149)(10,64,47,111,22,83,127,152)(11,71,56,114,23,90,136,155)(12,78,45,117,24,97,125,158)(13,65,54,120,25,84,134,141)(14,72,43,103,26,91,123,144)(15,79,52,106,27,98,132,147)(16,66,41,109,28,85,121,150)(17,73,50,112,29,92,130,153)(18,80,59,115,30,99,139,156)(19,67,48,118,31,86,128,159)(20,74,57,101,32,93,137,142), (61,100)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,99)(101,142)(102,143)(103,144)(104,145)(105,146)(106,147)(107,148)(108,149)(109,150)(110,151)(111,152)(112,153)(113,154)(114,155)(115,156)(116,157)(117,158)(118,159)(119,160)(120,141) );
G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(1,61,46,104,33,100,126,145),(2,68,55,107,34,87,135,148),(3,75,44,110,35,94,124,151),(4,62,53,113,36,81,133,154),(5,69,42,116,37,88,122,157),(6,76,51,119,38,95,131,160),(7,63,60,102,39,82,140,143),(8,70,49,105,40,89,129,146),(9,77,58,108,21,96,138,149),(10,64,47,111,22,83,127,152),(11,71,56,114,23,90,136,155),(12,78,45,117,24,97,125,158),(13,65,54,120,25,84,134,141),(14,72,43,103,26,91,123,144),(15,79,52,106,27,98,132,147),(16,66,41,109,28,85,121,150),(17,73,50,112,29,92,130,153),(18,80,59,115,30,99,139,156),(19,67,48,118,31,86,128,159),(20,74,57,101,32,93,137,142)], [(61,100),(62,81),(63,82),(64,83),(65,84),(66,85),(67,86),(68,87),(69,88),(70,89),(71,90),(72,91),(73,92),(74,93),(75,94),(76,95),(77,96),(78,97),(79,98),(80,99),(101,142),(102,143),(103,144),(104,145),(105,146),(106,147),(107,148),(108,149),(109,150),(110,151),(111,152),(112,153),(113,154),(114,155),(115,156),(116,157),(117,158),(118,159),(119,160),(120,141)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 7 | 2 | 0 | 0 | 0 | 0 |
| 16 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 9 | 9 | 3 |
| 0 | 0 | 32 | 0 | 39 | 4 |
| 0 | 0 | 0 | 0 | 28 | 28 |
| 0 | 0 | 0 | 0 | 13 | 32 |
| 0 | 35 | 0 | 0 | 0 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 14 | 31 | 17 |
| 0 | 0 | 37 | 33 | 28 | 0 |
| 0 | 0 | 13 | 8 | 0 | 28 |
| 0 | 0 | 4 | 28 | 3 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 29 | 20 |
| 0 | 0 | 0 | 1 | 40 | 36 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [7,16,0,0,0,0,2,34,0,0,0,0,0,0,28,32,0,0,0,0,9,0,0,0,0,0,9,39,28,13,0,0,3,4,28,32],[0,7,0,0,0,0,35,0,0,0,0,0,0,0,38,37,13,4,0,0,14,33,8,28,0,0,31,28,0,3,0,0,17,0,28,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,29,40,40,0,0,0,20,36,0,40] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 5 | 8A | ··· | 8H | 10A | ··· | 10G | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 4 | 20 | ··· | 20 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | M4(2) | M4(2) | F5 | C2×F5 | C2×F5 | C22.F5 | C4⋊F5 | D5⋊M4(2) |
| kernel | C20⋊8M4(2) | C20⋊C8 | Dic5⋊C8 | C2×C4×Dic5 | C2×C22.F5 | C4×Dic5 | C22×Dic5 | C22×C20 | C2×Dic5 | C2×Dic5 | Dic5 | C20 | C22×C4 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_{20}\rtimes_8M_{4(2)} % in TeX
G:=Group("C20:8M4(2)"); // GroupNames label
G:=SmallGroup(320,1096);
// by ID
G=gap.SmallGroup(320,1096);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,120,758,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^3,a*c=c*a,c*b*c=b^5>;
// generators/relations